In what follows, we are interested in studying the problem of smooth linearization of nonautonomous systems whose linear part admits a nonuniform polynomial dichotomy. We will consider systems with both discrete and continuous time. This problem consists of finding a time dependent smooth change of coordinates sending the solutions of the nonlinear system into the solutions of the linear one.
In our main results, we first formulate smooth linearization results for one-sided systems whose linear part admits a nonuniform exponential dichotomy, which are deduced from some existing results for two-sided systems [1]}, [2]}. Then, these results are applied to the setting of a nonuniform polynomial dichotomy to obtain smooth linearization results when the associated polynomial dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. Our techniques rely on the relationship between polynomial and exponential dichotomies via a suitable reparametrization of time. A key concept that is used in our results is given by the polynomial dichotomy spectrum and an important part of the paper is devoted to study the properties of this set. We emphasize that our results are in sharp contrast to most of the previous results in the literature which assumed that the linear part admits an (uniform/nonuniform) exponential dichotomy.
